The increment nature of relaxation rates, \(\Lambda \)  and \(\Delta \) , with temperature (Fig. REF  (b) and (c)), can be attributed to either fast fluctuations arising from electronic spins or the presence of a static or quasi-static magnetic field in the sample below the superconducting transition temperature. Considering the case of relaxation due to fluctuations, the field fluctuations perpendicular to an applied field can cause spin-flip transitions of the muon spin, which leads to an overall relaxation of the muon ensemble and a Curie-Weiss-like temperature dependence [1]}. Further, the slow time evolution of LF asymmetry spectra (Fig. REF  (a)), i.e., the low relaxation rate, indicates the decoupling of muon spin from the local magnetic field environment even with an applied magnetic field as low as 5 mT. This implies that the observed signal of increased relaxation rate with temperature is emerging from a dilute static or quasi-static magnetic field in the system, breaking time-reversal symmetry and excluding the possibility of any extrinsic magnetic impurity effects in Lu\(_3\) Os\(_4\) Ge\(_{13}\) . Such a change in \(\Lambda \)  has only been observed in a limited set of superconductors, including Sr\(_2\) RuO\(_4\)  [2]}, LaNiC\(_2\)  [3]}, SrPtAs [4]}, (Lu/Y/Sc)\(_5\) Rh\(_6\) Sn\(_{18}\)  [5]}, [6]}, [7]}, Ba\(_{1-x}\) K\(_x\) Fe\(_2\) As\(_2\)  [8]}, La\(_7\) X\(_3\)  (X = Rh,Ir,Pd,Ni) [9]}, [10]}, [11]}, [12]} series and recently in monochalcogenide ScS [13]}. Furthermore, the \(\Delta \)  relaxation channel, representing the correlated nuclear dipolar moments of the atoms, has a variation in the low-temperature region significantly below the superconducting transition temperature. The long-baseline value of \(\Delta \)  is most likely due to Lu atoms having the largest nuclear moment among the three constituent atoms. Besides, the trend in the secondary channel, \(\Delta \)  here, is also observed in La\(_7\) Ir\(_3\) , La\(_7\) Pd\(_3\) , La\(_7\) Rh\(_3\)  [9]}, [11]}, [12]} and Pr\(_{1-x}\) La\(_{x}\) Pt\(_4\) Ge\(_{12}\) , Pr(Os\(_{1-x}\) Ru\(_x\) )\(_4\) Sb\(_{12}\) , Pr\(_{1-y}\) La\(_y\) Os\(_4\) Sb\(_{12}\)  [17]}, [18]}. The proposed reason for the increment in \(\Delta \)  is nuclear spin fluctuations, though the net change observed in \(\Delta \)  for the aforementioned materials is much lower than the value detected for our single-crystalline Lu\(_3\) Os\(_4\) Ge\(_{13}\) .
